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Center for Nonlinear Studies |
Monte Carlo methods approximate integrals by sample averages of integrand values. Quasi-Monte Carlo methods can provide accurate approximations to high dimensional integrals in terms of (weighted) sample means of the integrand values at well chosen sites. Such integrals arise in a variety of applications, including finance, statistical physics, sensitivity analysis, and Bayesian inference. The cubature error depends on properties of the integrand, and how well the sampling approximates the probability measure.
This tutorial highlights some of the major approaches to analyzing and reducing the cubature error that the conference presentations will develop in great detail. There are deterministic, randomized, and Bayesian ways of viewing cubature error, each making different assumptions about the integrand and the sampling measure. When the integrands lie in a Hilbert space or can be modeled as instances of a Gaussian process, the error analysis is particularly elegant.
Some strategies for increasing efficiency are described. We also show how the dimension of the integration domain may affect the error analysis. In some cases this leads to a curse of dimensionality, while in other cases the computational cost does not explode as the dimension becomes arbitrarily large. Finally, we discuss progress in developing data-based error bounds, which can be used to determine the sample size required to meet a desired error tolerance.
Slides from previous presentation of this talk are at http://mcqmc2016.stanford.edu/Hickernell-Fred.pdf
Host: James Hyman